How little is the question.
If 18+ I can recommend the Old Trip to Jerusalem. Arguably the oldest pub in the UK, great pub that's part black & white building and part cave. And coincidentally where I met my wife

.
If a little younger, there was the Robin Hood museum on Maid Marien Way, which was pretty old when last I went, but could be interesting for a young'un.
There's also Games Workshop headquarterson Lenton Lane (iirc). Now although it's a war gaming company, the inside of the HQ (looks like a warehouse from the outside) is decked out like a medieval castle. There's also a medieval tavern themed pub which (used to) do decent food. Again, not been in a while.
Finally, if you want to do the whole dog walking and ice cream thing, the Nottingham University campus (10 min bus's ride from the city centre). It has some nice parkland, and a nice outdoor playground, and a nice lake to walk round.
Not sure if any of that helps, however, I was there as a student, so my target venues and activities were slightly different to yours.